The filtering method of MEMS gyro signal based on sparse decomposition

Abstract

Traditional denoising methods for Micro-electromechanical Systems (MEMS) gyro signal are required to obtain a priori noise statistical properties, which result in poor denoising performance in MEMS gyro utilized in Micro-Inertial Measurement While Drilling (MWD), due to the unknown and complex noise characteristics in MWD. According to this problem, a kind of gyro signal denoising method based on sparse decomposition without a requirement of the priori noise characteristics, utilizing a newly designed atom dictionary, is proposed. Firstly, the MEMS gyro output differential equation is established on the basis of the physical mechanism of the MEMS gyro, then the real MEMS gyro output signal characteristics are analyzed according to the solution of the differential equation. Secondly, the characteristic wave atom most similar to the gyro output signal is designed. Finally, the gyro signal sparse decomposition denoising experiments based on the designed atom dictionary are conducted, compared with the wavelet threshold method and Kalman filter. The experiment results show that the proposed denoising method based on sparse decomposition utilizing the newly designed atom dictionary outperforms wavelet threshold method and Kalman filter in MEMS gyro signal processing of MWD, especially when the noise statistical properties of gyro signal are completely unknown.

Keywords

Measurement while drilling MEMS gyro sparse decomposition signal denoising

1. Introduction

In recent years, Micro-electromechanical Systems (MEMS), are becoming increasingly popular in the drilling industry, because of their low cost, small footprint, long service life, improving precision, impact resistance, and low power consumption. These advantages make Micro Inertial Measurement Unit (MIMU), based on MEMS triaxial accelerometer and triaxial gyroscope, perfect candidates for Measurement While Drilling (MWD) characterized by high temperatures and substantial vibrations and shocks [1, 2, 3, 4, 5, 6, 7, 8]. As the main surveying device of MIMU, the triaxial gyro is capable of providing continuous triaxial angular velocity measurement used to obtain the orientation [3, 9, 10]. However, the substantial irregular noises associated with the surveyed MEMS gyro signal render the MEMS gyro-based MWD method unsuitable for long-term continuous orientation estimation [11, 12, 13]. Therefore, denoising the surveyed gyro signal and extracting real gyro signal are of great importance to improve the orientation estimation accuracy.

Among existing MEMS gyro signal denoising algorithms, the wavelet denoising method [14, 15, 16], famous for excellent time-frequency resolution characteristics, aims at obtaining the wavelet coefficients over different scales utilizing Mallet algorithm and setting to zero the smaller wavelet coefficients, in comparison with the preset threshold coefficient. However the determination of preset threshold coefficient is required to obtain a priori statistical properties of signal noise [17, 18], which leads to poor performance in MEMS gyro signal denoising utilized in MWD.

Another popular class of MEMS gyro signal denoising method is based on the Kalman filter [11, 19, 20], the basic idea of which is to establish the MEMS gyro signal state and observation model with random interference and noise, and then using the linear minimum variance of the state covariance sequence as the optimization criterion to give the optimal estimate of the MEMS gyro signal. Under certain stochastic input assumptions, the Kalman filter acts as the optimal state estimator for a linear system [21, 22]. However the input assumptions are not always suitable for long-term continuous estimation of the MEMS gyro signal, because of the strong random input noise. Besides, the established MEMS gyro signal state and observation model are always inapplicable in long-term continuous filtering process in MWD [23, 24]. As a result, this Kalman filter-based method is unsuitable for MEMS gyro signal de-noising in MWD.

Traditional MEMS gyro signal denoising methods, as shown above, require to obtain a priori statistical properties of signal noise, which result in poor denoising performance in MWD, due to the unknown and complex noise characteristics of gyro signal in MWD. To resolve the above problems, a kind of MEMS gyro signal denoising method based on sparse decomposition is proposed, when the priori statistical properties of gyro signal noise are unknown while drilling.

From the perspective of sparse decomposition and sparse representation while drilling, the surveyed MEMS gyro signal consists of two components, namely the real gyro signal and noises composed of quantization noise, rate random walk, bias instability, angular random walk, rate ramp and random noise. These noises, characterized by random, uncorrelation and uncertain structures, are uncorrelated with the real gyro signal, while the real gyro signal is a sparse component of the surveyed MEMS gyro signal with a certain inherent feature. Therefore, we can design an over-complete atom dictionary D, atoms of which correlate well with the real gyro signal structures, then the real gyro signal can be faithfully recovered from the measured gyro signal while drilling, completing the MEMS gyro signal denoising process without prior information of the noise statistical characteristics.

2. Analysis of the characteristics of MEMS gyro signal for MWD

The structure of MWD MEMS gyro is shown in Fig. 1, and the working principle of which is described as follows: The mutual reversed-phase AC voltage is applied to the electrostatic comb driver in pairs, with electrostatic force generated at the same frequency as the AC voltage. Under the action of electrostatic force, electrostatic comb driver starts vibration in the input axis and achieves resonance state by means of a control circuit. According to the dynamics mechanism of the gyro, when the gyro senses angular velocity on sensing-axis (Z-axis), the gyro will apperceive the Coriolis inertial force Output-axis (Y-axis) modulated by vibration signal in Input-axis (X-axis), and the apperceived Coriolis inertial force is proportional to angular velocity in size. Under the action of the Coriolis inertial force, vibration is produced along Output-axis with the detection capacitance changed, and then the electrical signal on the output-axis is obtained by using micro-capacitance circuit to detect changes in capacitance; Finally, the Output-axis signal is demodulated by the vibration signal of the Input-axis, and then becomes the final gyro output signal proportional to the angular velocity via a low-pass filter.

Figure 1.

Structure sketch of MWD MEMS gyro.

For a clear understanding of the following derivation, we first define a notation that will be utilized throughout this Section. We refer to the dynamic frame A-XYZ attached to the moving mass, and define the inertial frame o-xyz fixed in inertial space, as shown in Fig. 1. $\omega_{x}$ , $\omega_{y}$ and $\omega_{z}$ denote tri-axis angular velocity about the $x$ , $y$ and $z$ axis, respectively, of the dynamic frame relative to the inertial frame; the tri-axis displacement of the X, Y and Z axis of the moving mass centroid about the dynamic frame at time $t$ are termed $x(t)$ , $y(t)$ and $z(t)$ respectively; ${x}^{\prime}(t)$ , ${y}^{\prime}(t)$ and ${z}^{\prime}(t)$ correspond to the tri-axis velocity of the X, Y and Z axis of the moving mass centroid about the dynamic frame at time $t$ respectively; ${x}^{\prime\prime}(t)$ , ${y}^{\prime\prime}(t)$ and ${z}^{\prime\prime}(t)$ represent the tri-axis acceleration of the X, Y and Z axis of the moving mass centroid about the dynamic frame at time $t$ respectively.

When the dynamic frame A-XYZ rotates in the inertial frame o-xyz at an angular velocity $\Omega$ , the absolute acceleration of any point B in O-xyz can be expressed as:

$\displaystyle a_{B}=a_{A}+a_{B/A}+2\Omega\times V_{B/A}+\dot{\Omega}\times R_{% B/A}+\Omega\times(\Omega\times R_{B/A})$ (1)

where, $a_{A}$ is the acceleration in A-XYZ; $a_{B/A}$ , $R_{B/A}$ and $V_{B/A}$ define acceleration, position vector and velocity vector of point B relative to A-XYZ respectively; $\Omega$ and $\dot{\Omega}$ represent angular velocity and angular acceleration of A-XYZ with respect to O-xyz; $2\Omega\times V_{B/A}$ , $\dot{\Omega}\times R_{B/A}$ and $\Omega\times(\Omega\times R_{B/A})$ stand for Coriolis acceleration, tangential acceleration and centripetal acceleration of point B relative to O-xyz respectively.

Both the displacement and velocity of Z axis are much smaller than those of X axis and Y axis,because the stiffness of Z axis is much larger than that of X axis and Y axis in this MWD MEMS gyro, namely: $z(t)<<x(t)$ , $z(t)<<y(t)$ and ${z}^{\prime}(t)<<{x}^{\prime}(t)$ , ${z}^{\prime}(t)<<{y}^{\prime}(t)$ . When the origin of A-XYZ(A) coincides with the origin of o-xyz(O), $a_{A}=$ 0. On the basis of Newton second law and Eq. (1), the gyro input and output equations can be obtained respectively as follows:

$\displaystyle m_{x}{x}^{\prime\prime}(t)+D_{x}{x}^{\prime}(t)+\left[K_{x}-m_{x% }\left(\omega_{y}^{2}+\omega_{z}^{2}\right)\right]x(t)-2m_{x}\omega_{z}{y}^{% \prime}(t)+m_{x}(\omega_{x}\omega_{y}-\dot{\omega}_{z})y(t)$ (2) $\displaystyle\quad=F_{e}+M_{x}(t)$ $\displaystyle m_{y}{y}^{\prime\prime}(t)+D_{y}{y}^{\prime}(t)+\left[K_{y}-m_{x% }\left(\omega_{x}^{2}+\omega_{z}^{2}\right)\right]y(t)+2m_{y}\omega_{z}{x}^{% \prime}(t)+m_{y}(\omega_{x}\omega_{z}+\dot{\omega}_{z})x(t)$ (3) $\displaystyle\quad=M_{y}(t)$

where $m_{x}$ , $m_{y}$ , $D_{x}$ , $D_{y}$ , $K_{x}$ , $K_{y}$ and $M_{x}$ , $M_{y}$ represent the quality, damping coefficient, elastic coefficient and disturbing force of the input and output axis respectively. $F_{e}=f_{d}\sin\omega t$ is electrostatic driving force along the X-axis ( $f_{d}$ is driving amplitude; $\omega$ is driving frequency).

For better understanding the nature of the MWD MEMS gyro, the following derivation is to achieve a reasonable simplification of Eqs (2) and (2). Assuming that: (1) the moving mass is only applied to electrostatic force in X-axis without disturbing force; (2) the input angular velocity is much less than the natural frequency $\omega_{cx}$ , $\omega_{cy}$ of the X and Y axis (namely $\omega_{x}<<\omega_{cx}$ and $\omega_{y}<<\omega_{cy})$ ; (3) Only Z-axis senses the input angular velocity, neglecting $\omega_{x}^{2}$ , $\omega_{y}^{2}$ , $\omega_{x}\omega_{y}$ . Then Eqs (2) and (2) can be simplified as:

$\displaystyle m_{x}{x}^{\prime\prime}(t)+D_{x}{x}^{\prime}(t)+\left(K_{x}-m_{x% }\omega_{z}^{2}\right)x(t)-2m_{x}\omega_{z}{y}^{\prime}(t)=F_{e}$ (4) $\displaystyle m_{y}{y}^{\prime\prime}(t)+D_{y}{y}^{\prime}(t)+\left(K_{y}-m_{y% }\omega_{z}^{2}\right)y(t)+2m_{y}\omega_{z}{x}^{\prime}(t)=0$ (5)

Equations (4) and (5) describe the general form of single-axis MWD MEMS gyro input and output differential equations.

2.1 Analysis of MWD MEMS gyro output signal characteristic

The particular solution of the output equation is obtained from the Eqs (4) and (5):

$\displaystyle y_{p}(t)=2\omega_{z}f_{d}[A_{1}\sin(\omega t+\phi_{y})+A_{2}\cos% (\omega t+\phi_{y})]/(A_{1}^{2}+A_{2}^{2})$ (6)

where,

$\displaystyle A_{1}=m_{y}\omega^{4}-\frac{m_{y}D_{x}+D_{x}D_{y}+m_{y}K_{x}}{m_% {x}}\omega^{2}+\frac{D_{y}K_{x}+D_{x}K_{x}}{m_{x}},A_{2}=D_{y}\omega^{3}-\frac% {D_{y}K_{x}+D_{x}K_{x}}{m_{x}}.$

The equation is given by the output Eq. (5):

$\displaystyle{s}^{\prime\prime}+2\zeta_{y}\omega_{cy}{s}^{\prime}+\omega_{cy}^% {2}s=0.$ (7)

where, $\omega_{cy}^{2}=(K_{y}-m_{x}\omega_{z}^{2})/m_{y}$ , $\zeta_{y}=D_{y}/2m_{y}\omega_{cy}$ is damping ratio of the output signal equation.

The root of Eq. (5) is computed by the Eq. (7):

$\displaystyle s_{y}=-\zeta_{y}\omega_{cy}\pm\omega_{cy}\sqrt{\zeta_{y}^{2}-1}$ (8)

The damping can be divided into three cases according to the damping ratio. Case(1): over-damping, when $\zeta_{y}>1$ ; Case(2): critical-damping, when $\zeta_{y}=$ 1; Case(3): under-damping, when $\zeta_{y}<$ 1. When the damping ratio $\zeta_{y}\geqslant 1$ , the gyro output signal reaches stable state consuming a very long time; When $\zeta_{y}<1$ , The output signal decays exponentially with time, and the decay rate depends on the attenuation coefficient $\zeta_{y}\omega_{cy}$ . In order to obtain a shorter stabilization time in MWD, the damping ratio is often chosen as 0.6 [25].

The general solution of the output equation on condition of underdamping is given by Eqs (4), (5) and (8):

$\displaystyle y_{g}(t)=e^{-\zeta_{y}\omega_{cy}t}\left[A_{3}\cos\left(\omega_{% cy}\sqrt{1-\zeta_{y}^{2}}\cdot t+\varphi_{0}\right)+A_{4}\sin\left(\omega_{cy}% \sqrt{1-\zeta_{y}^{2}}\cdot t+\varphi_{0}\right)\right]$ (9)

where $\varphi_{0}$ is determined by the gyro structure constant, $A_{3}$ and $A_{4}$ can be calculated from $y_{g}(0)$ and $y_{g}^{\prime}(0)$ .

According to Eqs (6) and (9), the full solution of the MWD MEMS gyro output signal is:

$\displaystyle y(t)=y_{p}(t)+y_{g}(t)=B_{1}\omega_{z}\sin(\omega t+\phi_{y}+% \theta_{1})+B_{2}e^{-\zeta_{y}\omega_{cy}t}\sin\left(\omega_{cy}\sqrt{1-\zeta_% {y}^{2}}\cdot t+\phi_{0}+\theta_{2}\right)$ (10)

where, $B_{1}=2f_{d}(A_{1}^{2}+A_{2}^{2})^{-1/2}$ , $\theta_{1}=\tan^{-1}(A_{2}/A_{1})$ , $B_{2}=\sqrt{A_{3}^{2}+A_{4}^{2}}$ , $\theta_{2}=\tan^{-1}(A_{4}/A_{3})$ .

As can be readily seen from Eq. (10), the real MWD MEMS gyro output signal is composed of two components, and the first part is the particular solution with the driving frequency $\omega$ , namely the steady-state solution of the output signal $y(t)$ ; The other part is the transient solution, decaying exponentially with time.

2.2 Sparse analysis of MEMS gyro signal

In MWD, the surveyed gyro signal is often mixed with quantization noise, rate random walk, bias instability, angular random walk, rate ramp and random noise. From the perspective of signal generation mechanism, each of the components in gyro output signal must be caused by a corresponding drive, on account of which, the conclusion can be reached that: the MEMS gyro signal is sparse.

According to the sparse decomposition theory, the surveyed gyro signal in MWD consists of two parts: real gyro signal and noise. The real gyro signal is the sparse component of the surveyed gyro signal and characterized by the particular structure, as shown in Eq. (10), so that only some basis functions associated with the real gyro signal structure are sufficient to represent the true MEMS gyro signal, and the rest of the surveyed gyro signal can be viewed as the corresponding ‘noise’. Then the surveyed MEMS gyro signal can be expressed as a kind of combination of expansion functions. Suppose that $x(n)$ is a length-N gyro signal measured in MWD, it can be written:

$\displaystyle x(n)=\sum\limits_{m=0}^{M-1}{\alpha_{m}\psi_{m}(n)}+\varepsilon$ (11)

where $\psi_{m}(n)$ represents a certain atom related to the real gyro signal structure, $\alpha_{m}$ stands for an expansion coefficient over $\psi_{m}(n)$ , $\varepsilon$ denotes the noises in the real gyro signal.

Equation (11) can be rewritten in the following matrix form:

$\displaystyle x=D\alpha+\varepsilon$ (12)

where $\alpha$ is the expansion coefficient, which describes the similarity degree between the signal and the expansion functions; $D=\{\psi_{m}(n)\}_{m\in Z}$ defines as a $N\times M$ atoms matrix, which is called an atom dictionary; $D\alpha$ represents the real gyro signal; $\varepsilon$ denotes the noises.

3. Atom dictionary design

The sparse decomposition framework,as shown in Eq. (12) , is attractive as it implies that the real gyro signal can be faithfully recovered from the measured gyro signal in MWD, without prior information of the noise statistical characteristics and using $K<<N$ atoms in D. However, one challenging question in the sparse decomposition framework is how to design a suitable dictionary D. This Section discusses the atom dictionary D requirements.

3.1 Principle of atom dictionary design

As shown above, decomposing the noisy MEMS gyro signal over a family of atoms well localized both in time and frequency performs as an effective denoising method. Such atoms are called time-frequency atoms. We suppose that a single window function $g(t)\in L^{2}(R)(L^{2}(R)$ is the Hilbert space of complex valued functions), and $g(t)$ meets the following conditions simultaneously: (1) $g(t)$ is continuously differentiable; (2) $O(1/(t^{2}+1))$ ; (3) $||g||=1$ ; (4) $\int{g(t)dt\neq}$ 0s and $g(0)\neq$ 0, then $g(t)$ is referred to as a general family of time-frequency atoms. These conditions ensure the atoms are rich in local features and a fast attenuation speed. Then a general over-complete dictionary D of time-frequency atoms can be generated by scaling, translating and modulating.

3.2 MEMS gyro atom dictionary design

The essence of signal sparse decomposition is to obtain the original signal reconstruction accurately utilizing as few atoms as possible from the time-frequency atom dictionary D. If the designed time-frequency atoms can not correlate well with the signal structures, the finished decomposition is not necessarily complete signal sparse representation. Besides, the signal is subdecomposed into several elements and its information is diluted.

Equation (10) suggests that the real MEMS gyro output signal in MWD consists of periodic and exponential decay elements, and other existing components in the measured MEMS gyro output signal should be noises. Thus, exponential decay sine waves atoms can be selected to construct the time-frequency atom dictionary utilized in the sparse decomposition of the noisy MEMS gyro output signal, because of its good correlation with the two particular elements of a real MEMS gyro output signal.

The following function $\psi_{m}(t)$ is selected as the time-frequency atom dictionary D of the gyro signal sparse decomposition:

$\displaystyle\psi_{m}(t)=\frac{1}{\sqrt{c}}g\left({\frac{t-u}{s}}\right)\sin(% \omega t+\varphi)$ (13)

where $g(t)=e^{-\pi t}$ is a window function, $1/\sqrt{c}$ , $s$ , $u$ , $\omega$ and $\varphi$ define the normalized coefficient (ensuring the norm of $g(t)$ ), the expansion factor (scale factor), the translation factor (displacement factor), the atomic frequency and the atomic phase respectively.

According to Eq. (13), an atom is determined by $c,s,u,\omega$ and $\varphi$ , then a set of different $c,s,u,\omega$ and $\varphi$ yield an over-complete dictionary of time-frequency atoms D, where D represents $\{\psi_{m}|m=1,2,\ldots,p\}$ , p is the number of atoms in the over-complete dictionary.

The advantage of choosing Eq. (13) as an atom wave is its similarity to the transient response of the gyro output signal, so it is very suitable for extracting the attenuation waveform element in the gyro signal. Figure 2 shows a 512 points discrete atomic time domain waveform according to Eq. (13), where $c,s,u,\omega$ and $\varphi$ were set to $1/\sqrt{c}=1/\sqrt{400}$ , $s=$ 400, $u=$ 200, $\omega=$ 18 Hz and $\varphi=$ 0 respectively.

Figure 2.

A constructed atom curve.

In addition, when the scale factor $s$ is larger enough, the dictionary atoms degenerate into a standard sine wave function, as shown in Fig. 3. It also has the ability to extract the stable signal components characterized by a periodic sine wave. So the dictionary D, based on Eq. (13), takes into consideration both stationary and non-stationary characteristics of the real gyro signal.

Figure 3.

Constructed atom curves of different $s$ .

4. MEMS gyro signal sparse decomposition denoising

The general issue behind MEMS gyro signal sparse decomposition is to choose the atoms best adaptive to the structures of the signal to be decomposed, among all the time-frequency atoms of a large dictionary. In this Section, we introduce an algorithm called matching pursuit to decompose the surveyed MEMS gyro signal into a linear expansion of atoms belonging to the redundant dictionary D. A matching pursuit is a greedy algorithm that chooses an atom correlating well with the real gyro signal structures at each iteration.

According to Section 2.2, the real MEMS gyro signal behaves as a finite dimensional signal in the base space and is with sparsity, which indicates it can be efficiently represented by an appropriate countable subset of atoms from D.

Therefore, the reconstructed gyro signal can be written as:

$\displaystyle x_{r}=\sum\limits_{k=0}^{K-1}{\alpha_{k}g_{r_{k}}}$ (14)

where $x_{r}$ corresponds to the signal faithfully recovered from the measured gyro signal, $\alpha_{k}$ is sparse coefficient and $g_{r_{k}}$ represents the selected atoms.

As the dictionary D is very redundant, the choice of the sparse coefficient $\alpha_{k}$ is not unique, then the search for the optimal atom $g_{r_{k}}$ matching best the original measured gyro signal $x$ or its residues $x_{k}$ can be carried further by the matching pursuit. In the $k$ -th iteration:

$\displaystyle\alpha_{k}=<x_{k},\psi_{k}>=\max|<x_{k},\psi_{m}>|$ (15)

where $x_{k}$ and $\alpha_{k}$ represent the k-th residual signal of the surveyed MEMS gyro and the sparse coefficient respectively; $\psi_{m}$ and $\psi_{k}$ represent any one atom set and the best one adapted to $x_{k}$ from the dictionary D.

Then the iteration process is performed as follows:

Step(1): Initialization, let $x_{0}=x$ , $k=1$ ; Set up the iteration number as $K$ ;

Step(2): Figure out $\alpha_{k}=<x_{k},\psi_{k}>=\max|<x_{k},\psi_{m}>|$ ;

Step(3): Update the new residual signal $x_{k}$ and new approximation $x_{r}$ as follows:

$\displaystyle\left\{\begin{array}[]{l}x_{k}=x_{k-1}-\alpha_{k-1}\psi_{k-1}\\ x_{r}=\sum\limits_{i=0}^{k-1}{\alpha_{i}\psi_{i}}\\ \end{array}\right.$

Step(4): Update $k=k+1$ , if $k\leqslant K$ then repeat steps(2) and (3); Else end and output $x_{r}$ .

5. MWD MEMS gyro signal denoising experiment

The MEMS gyro signal was obtained as follows:

Step(1): The MEMS gyro was fixed on the turntable;

Step(2): Start the MEMS gyro acquisition system;

Step(3): Let the turntable rotate continuously around the vertical axis at 10( ${}^{\circ}$ /s) for two hours, meanwhile record both the turntable and the MEMS gyro data at a sampling interval of 0.02 s;

Step(4): Make the turntable swing continuously around the horizontal axis at 5 ${}^{\circ}$ /1 Hz for two hours, meanwhile record both the turntable and the MEMS gyro data at a sampling interval of 0.02 s.

5.1 MEMS gyro signal sparse decomposition experiment

From MEMS gyro output signal characteristics in Section 2.1, we can obviously learn that the real MEMS gyro signal is mainly composed of two structural types: periodic sinusoidal component and exponential attenuation component. Turntable swing experiment was designed as shown in Section 5.1 to check whether Eq. (10) can effectively characterize the real MEMS gyro signal and whether the Matching Pursuits (MP) sparse decomposition utilizing the designed atom dictionary based on Eq. (13) can effectively extract the real gyro signal from the noised MEMS gyro signal.

Considering that the angular velocity data collected from the turntable itself has much higher accuracy than that of MEMS gyro on measurement precision, the angular velocity measured by the turntable is approximately taken as the real angular velocity. Meanwhile, the installation error between MEMS gyro and turntable can be neglected. Then MP sparse decomposition based on the newly constructed atom dictionary of Eq. (13) is adopted to denoise the MEMS gyro signal, with the MP iteration times of s 15.

The angular velocity data collected from the turntable and the MEMS gyro are shown in Fig. 4a and b respectively. The denoised MEMS gyro signal utilizing MP sparse decomposition based on the newly designed atom dictionary of Eq. (13) and the residual MEMS gyro signal after sparse extraction are described in Fig. 4c and d respectively.

Figure 4.

Gyro signal sparse decomposition and reconstruction.

As can be readily seen in Fig. 4b, the obtained MEMS gyro signal, which is featured with periodic sinusoidal and exponential attenuation components, involves substantial noises in comparison with the angular velocity measured by the turntable in Fig. 4a. Comparing Fig. 4a and c, we can find clearly that the MP sparse decomposition based on the newly constructed atom dictionary of Eq. (13) achieves perfect approximation to the real MEMS gyro signal; Comparing Fig. 4a–c, lots of noises in the MEMS gyro output signal are removed, after sparse decomposition and reconstruction of the measured MEMS gyro signal, and the removed noise is shown in Fig. 4d.

5.2 Comparison of gyro signal denoising experiment

In order to comprehensively compare the denoising performance of MP sparse decomposition based on the newly constructed atom dictionary of Eq. (13), a total of 3 sets of MEMS gyro signal denoising methods are implemented, corresponding to the wavelet threshold, Kalman filter and sparse decomposition denoising utilizing the designed atom dictionary respectively. The 3 methods’ adjustable parameters are set to as follows: (1) Wavelet threshold denoising experiment: the Db4 orthogonal wavelet base is selected to perform the 5 scale wavelet decomposition of the MEMS gyro output signal, and the ‘minimax’ soft threshold is used to filter and reconstruct the MEMS gyro signal; (2) Kalman filter experiment: the values of coefficients are set to $P_{0}=$ 0.0032, $Q=$ 0.1, $R=$ 1; (3) MP sparse decomposition denoising experiment: the MP iteration number is set to 15.

The experiments are repeated 5 times, with each time using different length-10000 data from Section 5.1, to reduce accidental errors. A certain trial is shown in Fig. 5. The dynamic Signal-to-Noise Ratio(SNR) and Root Mean Square Error (RMSE) values of wavelet threshold, Kalman filter, and sparse decomposition utilizing the designed atom dictionary are computed respectively. The results are summarised in Table 1, where each value stands for the mean of all 5 experiments.

Table 1
Comparison of SNR/RMSE

Evaluation index	Db4 wavelet	Kalman filter	Sparse decomposition
SNR	61.5725	51.1065	65.6166
RMSE	0.0083	0.0278	0.0052

Figure 5.

Comparison of gyro angular velocity denoised by Db4 wavelet, Kalman filter and Sparse decomposition.

Comparing the results of the Db4 wavelet threshold method, Kalman filter and MP sparse decomposition denoising utilizing the designed atom dictionary, MP sparse decomposition denoising method utilizing the designed atom dictionary is superior to the other two methods in denoising performance quantized by SNR and RMSE.

6. Summary

In this paper, a kind of denoising method based on sparse decomposition in MEMS gyro utilized in MWD is proposed. This denoising method is most featured of not requiring information of noise statistical properties of gyro signals as a prior. However, the real gyro signal can be faithfully recovered from the noisy gyro signal while drilling depending on that the utilized atom dictionary should correlate well with the real gyro signal structures. Therefore, the emphasis of this study lies in: (1) Analysing the structures of the real gyro signal based on theoretical derivation and experimental verification; (2) Designing atom dictionary based on the characteristic of the real gyro signal; (3) Checking the performance of the newly designed atom dictionary utilized in the gyro signal sparse decomposition denoising compared with the wavelet threshold method and Kalman filter. The experiment results show that the proposed sparse decomposition denoising method utilizing the newly designed atom dictionary outperforms wavelet threshold method and Kalman filter in MEMS gyro signal processing of MWD, especially when the noise statistical properties of gyro signal are completely unknown. The results suggest the newly designed atom dictionary is applicable, providing a reference for the further application of sparse decomposition in denoising MEMS gyro signal while drilling.

Footnotes

Acknowledgments

This work is financially supported by the project of National Natural Science Foundation (60834005) and the key project of natural science research in Anhui Universities (KJ2018A0873) and the key project of the supporting plan for excellent young talents in Anhui Universities (gxyqZD2019105).

References

Wang

Noureldin

Iqbal

et al., A reduced inertial sensor system based on MEMS for wellbore continuous surveying while horizontal drilling, IEEE Sensors Journal 18 (2018), 5662–5673.

Miller

Erdos

, The dawn of MEMS sensors for directional drilling, Journal of Petroleum Technology 70 (2018), 18–19.

Xue

Wei

and Sun

, Continuous measurement-while-drilling utilizing strap-down multi-model surveying system, IEEE Transactions on Instrumentation and Measurement 63 (2014), 650–657.

Yuan

and Lan

, Design and algorithm verification of a gyroscope-based inertial navigation system for small-diameter spaces in multilateral horizontal drilling applications, Micromachines 6 (2015), 1946–1970.

Liu

and Song

, A low-cost calibration strategy for measurement-while-drilling system, IEEE Transactions on Industrial Electronics 65 (2017), 3559–3567.

Song

Huang

Zhang

et al., A new measurement-while-drilling system based on inertial technology, Advances in Mechanical Engineering 10 (2018), 1–14.

Wang

Wei

Zhao

et al., A MEMS resonant accelerometer for low-frequency vibration detection, Sensors and Actuators A: Physical 283 (2018), 151–158.

Nez

Fradet

Laguillaumie

et al., Simple and efficient thermal calibration for MEMS gyroscopes, Medical Engineering & Physics 55 (2018), 60–67.

Straeten

Rajai

and Ahamed

M.J.

, Method and implementation of micro Inertial Measurement Unit (IMU) in sensing basketball dynamics, Sensors and Actuators A: Physical 293 (2019), 7–13.

10.

Wang

and Zhu

, The research of PNS based on micro inertial sensors, Measurement 90 (2016), 576–582.

11.

Jurkov

A.S.

Cloutier

and Pecht

, Experimental feasibility of the in-drilling alignment method for inertial navigation in measurement-while-drilling, IEEE Transactions on Instrumentation & Measurement 60 (2011), 1080–1090.

12.

Bryne

Rogne

Fossen

et al., Attitude and heave estimation for ships using MEMS-based inertial measurements, IFAC-PapersOnLine 49 (2016), 568–575.

13.

Chen

Aktakka

E.E.

and Woo

, On-chip capacitive sensing and tilting motion estimation of a micro-stage for in situ MEMS gyroscope calibration, Mechatronics 56 (2018), 242–253.

14.

Alam

and Rohac

, Adaptive data filtering of inertial sensors with variable bandwidth, Sensors 15 (2015), 3282–3298.

15.

Zou

et al., Evaluation of MEMS structures with directional characteristics based on FRAT and lifting wavelet, Procedia CIRP 27 (2015), 298–302.

16.

Bayer

F.M.

Kozakevicius

A.J.

and Cintra

R.J.

, An iterative wavelet threshold for signal denoising, Signal Processing 162 (2019), 10–20.

17.

Liu

Chen

and Fan

, Wavelet Transform for Fresnel-transformed mother wavelets, Chinese Physics B 20 (2011), 57–62.

18.

and Wang

, Development of a novel knock characteristic detection method for gasoline engines based on wavelet-denoising and EMD decomposition, Mechanical Systems and Signal Processing 117 (2019), 517–536.

19.

Cai

Ding

et al., A noise reduction method for MEMS gyroscope based on direct modeling and kalman filter, IFAC-PapersOnLine 51 (2018), 172–176.

20.

Wang

and Wu

, In-flight initial alignment for small UAV MEMS-based navigation via adaptive unscented Kalman filtering approach, Aerospace Science and Technology 61 (2017), 73–84.

21.

Zheng

Liu

and Wei

, Adaptive Kalman filter based state of charge estimation algorithm for lithium-ion battery, Chinese Physics B 24 (2015), 581–587.

22.

Gao

Zhong

et al., Stochastic stability of the derivative unscented Kalman filter, Chinese Physics B 24 (2015), 64–73.

23.

Wang

Poscente

Filip

et al., Rotary in-drilling alignment using an autonomous MEMS-based inertial measurement unit for measurement while-drilling processes, IEEE Instrumentation Measurement Magazine 16 (2013), 26–34.

24.

Gao

Zhang

and Wang

, Research on initial alignment and self-calibration of rotary strapdown inertial navigation systems, Sensors 15 (2015), 3154–3171.

25.

Yang

and Li

, Physical modeling of some micro-mechanical gyroscope, Journal of Computational Information Systems 8 (2012), 143–150.

The filtering method of MEMS gyro signal based on sparse decomposition

Abstract

Keywords

1. Introduction

2. Analysis of the characteristics of MEMS gyro signal for MWD

3.1 Principle of atom dictionary design

3.2 MEMS gyro atom dictionary design

5.1 MEMS gyro signal sparse decomposition experiment

Table 1 Comparison of SNR/RMSE

Footnotes

Acknowledgments

References

Table 1
Comparison of SNR/RMSE